Calculation
κ = (ρ/h)·ln[(ρ+h/2)/(ρ−h/2)]−1; σ(y)=FN/A+Mb/(ρA)·[1+y/(κ(ρ+y))]
In a curved beam the zero-stress fibre shifts away from the centroidal line; the inner edge may be more highly stressed than a straight-beam formula suggests.
Gekrümmter Rechteckbalken nach Bernoulli · σ(y)=FN/A+Mb/(ρA)·[1+y/(κ(ρ+y))]
Calculate Inner and Outer Stress in a Curved Beam
In a curved beam the zero-stress fibre shifts away from the centroidal line; the inner edge may be more highly stressed than a straight-beam formula suggests.
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- Inner-edge normal stress σi — Signed normal stress at the fibre nearest the curvature centre; negative means compression. Select as result quantity.
- Outer-edge normal stress σa — Signed normal stress at the fibre away from the curvature centre; positive means tension. Select as result quantity.
- Centroidal curvature radius ρ — Distance from centre of curvature to the rectangular section centroidal line. Find it from inner radius plus half the radial section depth; must exceed h/2.
- Radial section depth h — Rectangular depth along the curvature radius between inner and outer edge; measure from drawing.
- Section width b — Constant rectangular width across the plane of curvature; with h it gives area A=b·h.
- Tangential axial force FN — Section force along the curved centroidal line; tension positive, compression negative. Determine from equilibrium for the load case.
- Bending moment Mb — Section bending moment in the curvature plane, positive when it increases curvature and puts the outer edge in tension. Find from loading and a section cut.
Example
For ρ = 100 mm, h = 100 mm, b = 20 mm, FN = 0 and Mb = 1,000 N·m, κ = ln 3−1 = 0.098612. The inner edge has −45.70 MPa compression and the outer edge +21.90 MPa tension.
Assumptions and limits
Planar circular beam with constant rectangular section, linear elastic material, plane sections and loads in the curvature plane. ρ>h/2 is required; transverse shear, local stress concentrations, plastic behaviour and three-dimensional effects are excluded. The result is nominal stress, not a strength verification.
Understand Edge stress in a curved rectangular beam
A curved hook or ring segment carries load differently from a straight beam. This calculator returns signed normal stress at the inner or outer edge of a curved rectangular section.
What does this quantity describe?
The unloaded centroidal line has radius ρ. For radial depth h and width b, section area is A=b·h. Curvature parameter κ=(ρ/h)·ln[(ρ+h/2)/(ρ−h/2)]−1 follows by integrating strain over the area. With tangential axial force FN and bending moment Mb, stress at distance y from the centroidal line is σ(y)=FN/A+Mb/(ρA)·[1+y/(κ(ρ+y))]. y=−h/2 is the inner edge and y=+h/2 the outer edge. The bending component is therefore nonlinear through the depth.
Formula and variables
κ = (ρ/h)·ln[(ρ+h/2)/(ρ−h/2)]−1; σ(y)=FN/A+Mb/(ρA)·[1+y/(κ(ρ+y))]
Area: A=b·hCurvature parameter: κ=(ρ/h)ln[(ρ+h/2)/(ρ−h/2)]−1Edge stress: σ(y)=FN/A+Mb/(ρA)[1+y/(κ(ρ+y))]Inner y=−h/2; outer y=+h/2
| Symbol / input | Meaning |
|---|---|
| Inner-edge normal stress σi | Signed normal stress at the fibre nearest the curvature centre; negative means compression. Select as result quantity. |
| Outer-edge normal stress σa | Signed normal stress at the fibre away from the curvature centre; positive means tension. Select as result quantity. |
| Centroidal curvature radius ρ | Distance from centre of curvature to the rectangular section centroidal line. Find it from inner radius plus half the radial section depth; must exceed h/2. |
| Radial section depth h | Rectangular depth along the curvature radius between inner and outer edge; measure from drawing. |
| Section width b | Constant rectangular width across the plane of curvature; with h it gives area A=b·h. |
| Tangential axial force FN | Section force along the curved centroidal line; tension positive, compression negative. Determine from equilibrium for the load case. |
| Bending moment Mb | Section bending moment in the curvature plane, positive when it increases curvature and puts the outer edge in tension. Find from loading and a section cut. |
Choose the inputs correctly
ρ is centroidal-line radius in mm, found from inner radius plus h/2. h is radial rectangular depth and b the width across the curvature plane, from the section drawing. ρ must exceed h/2 so inner radius is positive. FN is tangential section force in N, positive in tension and negative in compression. Mb is section bending moment in N·m, positive when curvature grows and the outer edge is in tension. Both section actions follow from load-case equilibrium. σi and σa are selectable nominal stresses in MPa; positive means tension and negative compression.
How to use the calculator
At the chosen section, determine FN and Mb using the stated signs. Enter rectangle dimensions and centroidal radius. Select inner edge first, then outer edge. Compare both signs and magnitudes with material data and any further local checks.
Worked example
For ρ=100 mm, h=100 mm, b=20 mm, FN=0 and Mb=1,000 N·m, area A=2,000 mm² and κ=ln 3−1=0.098612. The formula gives inner stress σi=−45.70 MPa and outer stress σa=+21.90 MPa. Here the inner magnitude is much greater.
Understand the result and units
Pure tensile axial force adds the same FN/A to both edges. Under a positive pure moment the inner edge is compressed and the outer edge stretched; curvature makes their magnitudes unequal. At large ρ the distribution approaches straight-beam bending.
The calculator converts ρ, h and b to m, FN to N and Mb to N·m. Area is m², κ dimensionless and σ calculated in Pa, displayed in MPa. All dimensions must refer to the same section.
Typical applications
Preliminary checks for crane hooks, ring segments and strongly curved frame parts with rectangular sections and planar loading.
Assumptions, limits and common mistakes
Circular beam with constant rectangular section, linear elastic material, plane sections and loading in the plane of curvature. Inner radius ρ−h/2 must be positive. Shear stress, notches, plastic stress distributions, lateral instability and three-dimensional connections are excluded. Nominal stress is not a full strength verification.
Common mistake: ρ is not inner radius; inner radius is ρ−h/2. Positive Mb puts the outer edge in tension. For a strongly curved section, do not treat the linear straight-beam formula as an exact edge stress.
Frequently asked questions
What is “Edge stress in a curved rectangular beam” used for?
Preliminary checks for crane hooks, ring segments and strongly curved frame parts with rectangular sections and planar loading.
Where do the input values come from?
ρ is centroidal-line radius in mm, found from inner radius plus h/2. h is radial rectangular depth and b the width across the curvature plane, from the section drawing. ρ must exceed h/2 so inner radius is positive. FN is tangential section force in N, positive in tension and negative in compression. Mb is section bending moment in N·m, positive when curvature grows and the outer edge is in tension. Both section actions follow from load-case equilibrium. σi and σa are selectable nominal stresses in MPa; positive means tension and negative compression.
What does the result not cover?
Circular beam with constant rectangular section, linear elastic material, plane sections and loading in the plane of curvature. Inner radius ρ−h/2 must be positive. Shear stress, notches, plastic stress distributions, lateral instability and three-dimensional connections are excluded. Nominal stress is not a full strength verification.